Problem 5

Let

\triangle OAB

have acute angle

AOB

. Through a point

M\ne O

, draw perpendiculars to

OA

and

OB

, with feet

P

and

Q

. Let

H

be the orthocenter of

\triangle OPQ

. Find the locus of

H

when

M

ranges over (a) the side

AB

; (b) the interior of

\triangle OAB

Step 1 of 4: Set up the fixed feet

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In plain words

The moving perpendiculars are parallel copies of two fixed altitudes.

BX\\perp OA,\\quad AY\\perp OB,\\quad MP\\parallel BX,\\quad MQ\\parallel AY.

Detailed analysis

Let $X$ be the foot from $B$ to $OA$ , and $Y$ the foot from $A$ to $OB$ . Since $MP\\perp OA$ and $MQ\\perp OB$ , we have $MP\\parallel BX$ and $MQ\\parallel AY$ .